Transcript ME 242 Chapter 13

Chapter 13
Dynamics
Chapter 3
Newton’s Law
NEWTON'S LAW OF INERTIA
A body, not acted on by any force, remains in
uniform motion.
NEWTON'S LAW OF MOTION
Moving an object with twice the mass will require
twice the force.
Force is proportional to the mass of an object and
to the acceleration (the change in velocity).
F=ma.
W = m*g
Fnet = F = T1 + T2 + T3 = 0
x- and y-components:
T1x = - T1 cos 37o = - 0.8 T1
T1y = T1 sin 37o = 0.6 T1
T2x = T2 cos 53o = 0.6 T2
T2y = T2 sin 53o = 0.8 T2
Solve for tension T1 and T2.
Fnet,x = F x = T1 x + T2 x + T3 x = 0
T1 x + T2 x + T3 x = 0
- 0.8 T1 + 0.6 T2 + 0 = 0
T1 = 0.75 T2
Dynamics
M1: up as positive:
Fnet = T - m1*g = m1 a1
M2: down as positive.
F
=
F
=
m
*g
T
=
m
a2
net
2
2
3. Constraint equation:
a1 = a2 = a
Equations
From previous:
T - m1*g = m1 a
 T = m1 g + m1 a
Previous for Mass 2:
m2*g - T = m2 a
Insert above expr. for T
m2 g - ( m1 g + m1 a ) = m2 a
( m2 - m1 ) g = ( m1 + m2 ) a
( m1 + m2 ) a = ( m2 - m1 ) g
a = ( m 2 - m 1 ) g / ( m1 + m 2 )
Rules
1. Free-Body Analysis, one for
each mass + Newton’s Law
2. Constraint equation(s):
Define connections.
You should have as many
equations as Unknowns.
COUNT!
3. Algebra:
Solve system of equations
for all unknowns
J
g
m
M*g*sinq*i
i
0 = 30
M*g
0
-M*g*cosq*j
Mass m rests on the 30
deg. Incline as shown.
Step 1: Free-Body
Analysis. Best
approach: use
coordinates tangential
and normal to the path
of motion as shown.
J
g
m
M*g*sinq*i
i
0 = 30
M*g
Mass m rests on the 30
deg. Incline as shown.
Step 1: Free-Body
Analysis.
Step 2: Apply Newton’s
Law in each Direction:
0
N
-M*g*cosq*j
(Forces_ x)  m* g *sinq *i  m* x
(Forces_ y)  N - m* g * cosq * j  0(static _ only)
Friction F = mk*N:
Another horizontal
reaction is added in
negative x-direction.
J
g
m
M*g*sinq*i
i
0 = 30
M*g
0
mk*N
N
-M*g*cosq*j
(Forces_ x)  (m* g *sinq  mk * N ) *i  m* x
(Forces_ y)  N - m* g * cosq * j  0(static _ only)
Midterm 1 :
Some suggestions
•Systematic work: it takes practice, lots
of it.
•Passive understanding is good, yet you
still must train yourself through active
practice.
•It’s all mathematical: Practice calculus
and analytical geometry!
Midterm 1:
Suggestions cont’d
•‘Deep Thinking 1’: Map the solution
path BEFORE starting the analysis.
‘Deep Thinking 2’: Select the laws
you will use.
•‘Deep Thinking 3’: Map out the Connections
between the laws that will lead to the answer.
Preparing for Exam 1
•Modeling: Free-Body Analysis
•Laws and Definitions: Laws of
Kinematics, Terms such as w, Coriolis
accel, Moving Frames of Ref.
•Seek to understand the concepts
•Practice problem solving.
Copying the homework gives you
an illusion and useless points
g
A
x
30 deg.
B
60 deg.
Multiple Masses A
and B move in i
and j-directions.
Steps:
1. Write Newton for
each Mass.
2. Constraint equation
connects both
masses. Here:
y
vA = - vB
Newton’s Law for Rotation
F
r
 m * ar
 F q  m * aq
fig_03_007
Step 1: Free-Body Analysis
Step 2: Newton in radial and tangential directions
Rotation Kinematics
Similar to translation:
w    * dt
q   w * dt
and
w * dw   * dq